gyrokinetic simulations for tokamaks and stellarators meeting... · gyrokinetic simulations for...

Gyrokinetic simulations for tokamaks and stellarators

R. Kleiber1, M. Borchardt1, M. Cole1, T. Feher2, R. Hatzky2,A. Konies1, A. Mishchenko1, J. Riemann1

1Max-Planck-Institut fur Plasmaphysik, D-17491 Greifswald, Germany

2Max-Planck-Institut fur Plasmaphysik, D-85748 Garching, Germany

February 26, 2015

This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the

European Union’s Horizon 2020 research and innovation programme under grant agreement number 633053. The views

and opinions expressed herein do not necessarily reflect those of the European Commission.

R. Kleiber 7th IAEA TM 1 / 43

Introduction

Interest in global (full-volume) gyrokinetic particle-in-cellsimulations for stellarators.

Two main areas of activity:

Microinstabilities and turbulenceMHD modes and their interaction with fast particles

Even linear electrostatic simulations for stellarators can becomedifficult due to high necessary resolution.

For electromagnetic perturbations already linear simulations in atokamak can become very difficult(especially in the MHD limit, small k⊥ρ)⇒ necessity of algorithm development

MHD modes provide an excellent testing ground for codes since avery high quality of numerics is required.


Overview

1 TheoryElectromagnetic gyrokinetic equationsMHD hybrid modelElectron fluid hybrid model

2 ResultsMHD hybrid modelElectron fluid modelFully electromagnetic gyrokineticsElectrostatic ITG in stellarators


Electromagnetic simulations: Hierarchy of models

Fully gyrokinetic simulations are very time-consuming and difficult.Develop simplified models: Sacrifice physics for gain in speed.⇒ EUTERPE, FLU-EUTERPE, CKA-EUTERPE

more

numerically

robust

and economical

more physically

complete

bulk plasma

GK fast ion

fluid electrons

(FLU

−EU

TER

PE)

power transfer

MHD

GK fast ions

(EUTERPE)

(CKA−EUTERPE)

bulk ions and electrons, fast ions

fluid bulk plasma

electromagnetic gyrokinetics for

GK bulk ions and fast ions


Full electromagnetic gyrokinetic model

All equations are derived from a Lagrangian via a variational principlein ~R, p‖, µ, α coordinates (consistency and conserved quantities)

L =∑

species

∫ f

[(q ~A+ p‖~b)· ~R+

m

qµα−

p2‖

2m− µB + q〈

p‖

mA‖ − Φ〉

]+

f0

[− q2

2m〈A‖〉2 +

m

2B2(∇⊥Φ)2

]dV dW − 1

2µ0

∫(∇⊥A‖)2dV

δB‖ neglected

〈·〉: gyro-averaging operator

dW = B∗‖ dp‖ dµ dα, dV =√g ds dϑ dϕ, B∗‖ = B + 1

qp‖~b·∇×~b

adiabatic electron model: m2B2 (∇⊥Φ)2 =⇒ e2

2kBTe0(Φ− Φ)2

neglect q〈Φ〉 for electrons


Cancellation problem

Electromagnetic simulations are hampered by the cancellation problem.

Moment of p‖ does not give a physical current.Ampere’s law in p‖ formulation (dc: collisionless skin depth)

− 1

β∇2⊥A‖ +

1

d2c

A‖ = j‖i + j‖e

1d2cA‖ 1

β∇2⊥A‖ (especially for MHD modes, since k⊥ρi ≈ 0)

j‖e contains adiabatic part j‖e,ad =∫ev‖

[eT f0v‖A‖

]d3v

Numerical cancellation of the two large terms 1d2cA‖ and j‖e,ad

Different numerical representations: matrix ⇔ particles

Cancellation problem mitigated by adjustable control variate method.[Hatzky et al. 2007]

Iterative method for determining a control variateAllows strong reduction in the number of necessary markers

Nevertheless, simulations require a very long time (small time step) orare still not feasible for certain parameters.


Pullback scheme I [Mishchenko et al. 2014]

Equations of motion in v‖ formulation (slab version for simplicity)

~R = v‖~b+1

B~b×∇

[φ− v‖A‖

]v‖ = − q

m

[∇‖φ+

∂A‖

∂t

]Use splitting A‖ = As

‖ +Ah‖ , introduce u‖ = v‖ + q

mAh‖ and

simplify equations by using the resulting freedom to postulate∂As‖

∂t +∇‖φ = 0

~R = u‖~b+1

B~b×∇

[φ− v‖(As

‖ +Ah‖)]− q

mAh‖~b

u‖ =q

mu‖∇‖Ah

‖

Field equations∂As‖

∂t+∇‖φ = 0 − 1

β∇2⊥(Ah

‖ +As‖) + SAh

‖ = j


Pullback scheme II

Kinetic equation:∂δf

∂t= − ~R·∇f0 −

q

mu‖∇‖Ah

‖∂f0

∂u‖

second term can be reduced by keeping Ah‖ small ⇒ idea of restarting

1 Start with Ah‖ = 0, i.e. u‖ = v‖

2 Integrate system in u‖-space: Ah‖ develops, leading to u‖ 6= v‖

3 Transform from u‖-space to v‖-space (use pullback)

δf(v‖) = δf(u‖) +q

mAh‖∂f0

∂u‖, v‖ = u‖ −

q

mAh‖

4 Keep the full solution by setting As‖ to As

‖ +Ah‖ and restart

Scheme allows for much larger time steps and enables simulations inparameter regimes not accessible before.


MHD hybrid model I

Use simplified way of describing the interaction of Alfven modeswith fast particles.

CKA code (A. Konies) solves the linearised reduced MHDeigenvalue problem.

Resulting mode structure and frequency used as a fixed input fora particle simulation ⇒ linear

Allow trajectories to react ⇒ nonlinear

Perturbative scheme

A

particle motion

CKA EUTERPE

eigenvalue problemMHD equation

Φ,

ω

gyrokintic equation


MHD hybrid model II

Time derivative of quasineutrality equation

∂

∂t∇·[Mn

B2∇⊥φ

]= −∇·

[−~b 1

µ0∇2⊥A‖ + j

(0)‖

(~b×~κB

A‖ −~b×∇A‖

B

)

+ p(1)bulk

(~b×~κB

+~b×∇BB2

)+ p

(1)‖,fast

~b×~κB

+ p(1)⊥,fast

~b×∇BB2

]CKA part, EUTERPE part

Simple closure for bulk pressure (neglect compressibility andanisotropy)

∂p(1)bulk

∂t= −

~b×∇φB·∇p(0)

bulk

MHD closure (vanishing E‖)

∂A‖

∂t= −~b·∇φ


MHD hybrid model: Amplitude equations I

Alfven modes are stable eigenmodes with frequency ω0

φ(~r, t) = φ0(~r) eiω0t

A‖(~r, t) = A‖0(~r) eiω0t

p(1)(~r, t) = p0(~r) eiω0t

Allow for complex time dependent amplitude

φ(~r, t) = φ(t) φ0(~r) eiω0t

A‖(~r, t) = A‖(t) A‖0(~r) eiω0t

p(1)(~r, t) = p(t) p0(~r) eiω0t


MHD hybrid model: Amplitude equations II

∂φ

∂t= iω0(A‖ − φ) + 2(γ − γd)φ

∂A‖

∂t= −iω0(A‖ − φ)

γ =T

2Wγd: ad-hoc damping

Wave-particle energy transfer:

T = −∫

dWdV f(1)fast

[1

B~b×(mv2

‖~κ+ µ∇B)·∇φ∗]

Wave energy:

W =

∫dV

Mn

B2|∇⊥φ|2

Linear model: Growth rate of the mode given by γNonlinear model: Use Re(φ),Re(A‖) for fast particle trajectories


Electron fluid model [Cole et al. 2014]

Linearised electron continuity equation (v‖ formulation)

∂ne

∂t+ n0

~B ·∇ue‖

B+B~vE×B ·∇

n0

B+ (∇×A‖~b)·∇

n0ue‖0

B+

n0(2~v∗ − ~vE×B)·∇BB

+∇×BB2

·(−1

e∇pe + n0∇φ

)= 0

with ~v∗ :=~b×∇peen0B

Ampere’s law and quasineutrality

ue‖ =1

en0

(ji‖ +

1

µ0∇2⊥A‖

)−∇·

(min0

eB2∇⊥φ

)= ni−ne

MHD (E‖ = 0) and pressure closure

∂A‖

∂t= −~b·∇φ ∂pe

∂t= −~vE×B ·∇pe0

More physics can be included by improving the closures.


Global gyrokinetic code EUTERPE

δf particle-in-cell code

Global simulation domain: full-volume

3D stellarator equilibria (from VMEC equilibrium code)

Multiple kinetic species (ions, electrons, fast ions/impurities)

Electrostatic/electromagnetic

Linear/nonlinear

Pitch-angle collision operator (⇒ neoclassics)

⇒ Code platform for implementation of different models(GYGLES: basically a reduced, linear, axisymmetric version of EUTERPE)


W7-X Toroidal Alfven eigenmode

W7-X equilibrium withβ ≈ 3%

Flat bulk plasma profilesnbulk = 1020 m−3

TAE mode found (CKA)

Alfven spectrum

0 0.2 0.4 0.6 0.8

norm. toroidal flux

0

0.2

0.4

0.6

0.8

ω/

ωA

TAE(m=6,7) mode

m = 6m = 7

m = 4

m = 5

m = 4m = 5

TAE mode

0 0.2 0.4 0.6 0.8 1

norm. toroidal flux

0

0.5

1

Re

(φ),

no

rma

lize

d

m= 7 n=-6

m= 6 n=-6

m= 5 n=-6

m= 4 n=-1 m= 5 n=-1

fast-ion density


W7-X TAE: Interaction with fast particles

Destabilise TAE mode by fastparticle interaction:Maxwellian fast particlesnfast profile fixed withnfast,0 = 1017 m−3

FLR effects important

Reaction on external radialelectric field( ~E0 = Es∇s, Es = const.)

ME =1

vth〈|~E0× ~BB2

|〉

0 5e+05 1e+06 1.5e+06 2e+06

Tf / eV

0

500

1000

1500

2000

2500

3000

γ /

(1

/s)

no FLRwith FLR

-0.012 -0.008 -0.004 0 0.004 0.008 0.012

ME

1500

2000

2500

3000

γ / (

1/s

)

Ion root

Electron root

[Mishchenko et al. 2014]


W7-X TAE: Interaction with ICRH fast particles

Fast ions from ICRH

Very simple model for ICRH:anisotropic Maxwellianpower deposition localised inspace

Less-pronounced finiteorbit-width effects

Strong influence of anisotropyon growth rate (more/lessparallel temperature)

0 0.2 0.4 0.6 0.8 1

norm. toroidal flux

0

0.5

1

1.5

2

Re

φ, n

orm

alize

d

m= 7 n=-6 m= 6 n=-6 m= 5 n=-6 m= 4 n=-1 m= 5 n=-1

min

. perp

. temp.

ICRH

minority density

0 500 1000 1500 2000

Tmax

(minority) / keV

0

1000

2000

3000

4000

γ / (

1/s

)

ICRH, no FLRICRH, with FLRMaxw., no FLRMaxw., with FLR


Nonlinear CKA-EUTERPE: amplitude development

ITPA benchmark case:circular A = 10 tokamak,(m,n) = (10/11,−6) TAE

Without artificial dampingmode amplitude does notsaturate.

Feature robust with respect totime step and particle number:no numerical artefact

Artificial damping necessary(γd ≈ 2.5 · 103 s−1) in order toreach saturation (damping ofsame order also used by othercodes).

no damping

WENDELSTEIN 7-X

Max-Planck-Institut für Plasmaphysik, EURATOM Association

potential does not saturate

0 200 400 600 800 1000 1200 1400−40

−35

−30

−25

−20

−15

−10

−5

0All modes

t, [103 Ω−1]

log(Φ

)

0 200 400 600 800 1000 1200 14000

0.05

0.1

0.15

0.2

0.25Selected modes

t, [103 Ω−1]

|Φ|

A. Könies, R. Kleiber, T. Fehér, M. Borchardt, R. Hatzky, A. Mishchenko · Max-Planck-Institut für Plasmaphysik

damping

WENDELSTEIN 7-X

Max-Planck-Institut für Plasmaphysik, EURATOM Association

saturated electrostatic potential

0 100 200 300 400 500 600 700−35

−30

−25

−20

−15

−10

−5

0All modes

t, [103 Ω−1]

log(Φ

)

0 100 200 300 400 500 600 7000

0.01

0.02

0.03

0.04

0.05

0.06Selected modes

t, [103 Ω−1]

|Φ|

damping value (GYGLES): γdG = 3.9 · 103s−1

damping value (LIGKA- P. Lauber): γdL = 2.3 · 103s−1

damping value set: γd = 2.5 · 103s−1

A. Könies, R. Kleiber, T. Fehér, M. Borchardt, R. Hatzky, A. Mishchenko · Max-Planck-Institut für PlasmaphysikR. Kleiber 7th IAEA TM 22 / 43

Nonlinear CKA-EUTERPE

For small growth-rates theory(Berk-Breizmann model)predicts relationship betweensaturation amplitude andgrowth rate: δB/B ∼ γ2

eff

Scaling confirmed by othercodes

Symmetric frequency chirpingobserved.

1000 10000 1e+05

γeff

/s-1

0.001

0.01

0.1

1

10

δB

rad/B

0 /

10

-3

CKA-EUTERPE γd= 1.05.10

4 s

-1


4 s

-1


3 s

-1

200 400 600 800 1000 1200 1400

−1.9

−1.8

−1.7

−1.6

−1.5

−1.4

−1.3

−1.2

−1.1

−1

x 10−3

t, [103 Ω

−1]

ω / Ω

*


FLU-EUTERPE benchmarks

Internal kink in a screw pinch.Scan over position of q = 1 surface.

0.5 0.6 0.7 0.8

rc / r

a

10000

20000

30000

40000

γ,

ra

d/s

MHD, shooting methodGK PIC (T

i = 5000 eV)

FLU-EUTERPE (Ti = 5000 eV)

ITPA benchmark: EUTERPE, FLU-EUTERPE and CKA-EUTERPE

0 2e+05 4e+05 6e+05 8e+05

Tf [eV] (n

0f fixed)

0.36

0.38

0.4

0.42

0.44

0.46

ω, x

10

6 r

ad/s gap

continuum

continuum

0 2e+05 4e+05 6e+05 8e+05

Tf [eV] (n

0f fixed)

0

10000

20000

30000

γ,

rad

/s

EUTERPEFLU-EUTERPECKA-EUTERPE


Internal tokamak kink mode

Finite ∇T used to destabilise themode.

In fluid limit (no gyrokinetic ions),direct comparison possible withMHD code CKA:γFLU = 1.29× 106 s−1

γCKA = 1.27× 106 s−1

Strong stabilisation withJET/ITER-like aspect ratio andelongation.

Further stabilisation with inclusionof bulk ion gyrokinetic effects.

0 0.2 0.4 0.6 0.8 1r, m

0e+00

2e-04

4e-04

6e-04

8e-04

Ele

ctro

stat

ic p

ote

nti

al

MHD m=0MHD m=1MHD m=2 GK PIC m=0GK PIC m=1GK PIC m=2

0 2.5 5 7.5 10Aspect ratio

0

2.5e+05

5e+05

7.5e+05

1e+06

1.25e+06

1.5e+06

1.75e+06

2e+06

rad/s

Bulk fluid (γ)Kinetic ions (γ)Kinetic ions (ω)

[Cole et al. 2014]

Resistive layer physics lost but tokamak simulations become practical.R. Kleiber 7th IAEA TM 26 / 43

Fast particle effects: Linear fishbone

Maxwellian fast species (D),Tfast = 300 keV

Initial stabilisation of themode (perturbative effect)overcome by unstable branch.

Frequency jump with onset offast particle destabilisation.

Non-perturbative modestructure for larger fastparticle density.

0 0.01 0.02 0.03 0.04 0.05ρ

f = n

f/n

i

0

50000

1e+05

1.5e+05

2e+05

2.5e+05

Gro

wth

rat

e (r

ad/s

)

Growth rateFrequency

ρf = 0 ρf = 0.01 ρf = 0.04


Limits of fluid models

For ITPA case TAE had nocontinuum interaction.

Use steeper q-profile⇒ continuum interaction.

MHD mode structure stronglymodified by interaction withKAW. [Cole et al. 2015]

Change of mode structure notcaptured by CKA-EUTERPE.

FLU-EUTERPE:follows change in mode structurebut break-down of fluid closure ?

frequency

0 0.2 0.4 0.6 0.8 1

sqrt norm tor. flux

0

2e+05

4e+05

6e+05

8e+05

ω , rad/s

m =

10

m =

10

m =

11

m =

11

TAE (MHD)

TAE (kin, Ti=9 keV)

m =

12

m =

13m

= 9

m =

14

m =

10

growth rate

2e+05 3e+05 4e+05 5e+05 6e+05 7e+05T

f eV

0

10000

20000

30000

40000

50000

γ, ra

d/s

EUTERPE, mixed variablesEUTERPE, conventional schemeFLU-EUTERPE (GK bulk, fast ions)

FLU-EUTERPE (GK fast ions)

CKA-EUTERPE


Pull-back scheme: simulations for stellarator

Electromagnetic ITG for LHD-like configuration(βeq = 1.5%, βe = 0.85%)

Standard δf scheme:

Numerical instabilityquickly develops

Wrong mode structure

New scheme:

Simulation stays stable

Clean mode structure

[Mishchenko et al. 2014]

0 0.2 0.4 0.6 0.8 1

norm. toroidal flux

-0.5

0

0.5

Re φ

−43−41

−39−37

−35−33

−31−29

−27

−83

−63

−43

−23

−3

17

37

0

0.2

0.4

0.6

0.8

1

mn

Φm

,n

0 0.2 0.4 0.6 0.8 1

norm. toroidal flux

0

0.2

0.4

0.6

0.8

1

Re φ

−43−41

−39−37

−35−33

−31−29

−27

−83

−63

−43

−23

−3

17

37

0

0.2

0.4

0.6

0.8

1

mn

Φm

,n


Simplified profiles for LT -Ln-scans

For global simulations profiles are a critical issue: One gains afunctional degree of freedom but looses flexibility to produceparameter sequences.

Simplified profiles:piece-wise lineard lnTi,e

ds andd lnni,e

ds

(s = Ψtor(r)Ψtor(0)).

Further simplification:keep equilibriumconfiguration fixed andchange parameters(LT, Ln) only in thesimulations.

0 0.2 0.4 0.6 0.8 10

1

2

temperature & density (η = 2.0)

sF

j / F *

F1=T

i (T

e=T

*)

F2=n

i,e

0 0.2 0.4 0.6 0.8 10

2

4

6

8

s

p / p

*

total pressure

EUTERPEVMEC

0 0.2 0.4 0.6 0.8 1

−4

−3

−2

−1

0logarithmic derivatives (η = 2.0)

s

d( ln

Fj )

/ ds

F1=T

i

F2=n

i,e


Quasi-realistic profiles for LT -Ln-scans

Quasi-realistic: simplified but derived from equilibrium(assuming finite residual pressure δp0)

p

p0= 1− 2s+ s2 +

δp0

p0

Ti,e

T0=

(1

2

p

p0

)1−χ, (Te = Ti)

ni,e

n0=

(1

2

p

p0

)χ, (ne = ni)

ηi =1− χχ

= const.

⇒ ηi-scan implies LT,n-scan

Advantage: not necessary torecalculate equilibrium for newparameter settings

0 0.2 0.4 0.6 0.8 10

1

2

temperature & density (η = 2.0)

s

Fj /

F *

F1=T

i,e

F2=n

i,e

0 0.2 0.4 0.6 0.8 10

2

4

6

8

s

p / p

*

total pressure

EUTERPEVMEC

0 0.2 0.4 0.6 0.8 1

−3

−2

−1

0logarithmic derivatives (η = 2.0)

s

d( ln

Fj )

/ ds

F1=T

i,e

F2=n

i,e


ITG stability diagrams (simplified profiles)

W7-X(β = 2%, T∗ = 1 keV)

0 1 2 3 4 5

0

0.05

0.1

0.15

0.2

0.25

0.3

W7−X ( β=2% ; T*=1keV )

ηi

γ (

vT/a

)

a/Ln = 1.41

a/Ln = 2.82

a/Ln = 4.23

a/Ln = 5.64

a/Ln = 7.05

LHD(β = 1.5%, T∗ = 1 keV, R0 = 3.75 m)

0 1 2 3 4 5

0

0.05

0.1

0.15

0.2

0.25

0.3

LHD ( β=1.5% ; T*=1keV )

ηi

γ (

vT/a

)

a/Ln = 1.41

a/Ln = 2.82

a/Ln = 4.23

a/Ln = 5.64

a/Ln = 7.05

Clear onset of linear ITG instability for ηi ≥ 1 observed.

Similar growth rates for W7-X and LHD.


W7-X: ITG mode pattern

W7-X (β = 2%, T∗ = 1 keV)

Electrostatic potential (absolute values) normalized to maximum

s = 0.5

a/LT = 1.41, a/Ln = 0.0, (m0, n0) = (210,−190)

ITG mode resides in region of unfavourable curvature.

High-shear region or ’helical edge’ avoided.


LHD: ITG mode pattern

LHD (β = 1.5%, T∗ = 1 keV, R0 = 3.75 m)

electrostatic potential (absolute values) normalized to maximum at s = const.

ηi = 2, s = 0.61, γ = 0.12 vT/a ηi = 1, s = 0.75, γ = 0.02 vT/a

Helical edge less pronounced than in W7-X


LT -Ln-scan for simplified profiles

LHD (β = 1.5%, T∗ = 1 keV)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8

gam

ma [v_T

/a]

a/L_T

a/L_n=0.00a/L_n=1.41a/L_n=2.82a/L_n=4.23a/L_n=5.64a/L_n=7.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8

gam

ma [v_T

/a]

a/L_n

a/L_t=0.50a/L_t=1.41a/L_t=2.82a/L_t=4.23a/L_t=5.64a/L_t=7.05


LT -Ln-scan for quasi-realistic profiles

LHD (β = 1.5%, T∗ = 1 keV)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8

gam

ma [v_T

/a]

a/L_T

a/L_n = 0.00a/L_n = 1.41a/L_n = 2.82a/L_n = 4.23a/L_n = 5.64a/L_n = 7.05a/L_p = 4.71

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8

gam

ma [v_T

/a]

a/L_n

a/L_t = 0.50a/L_t = 1.41a/L_t = 2.82a/L_t = 4.23a/L_t = 5.64a/L_t = 7.05

a/L_p = 4.71

Quasi-realistic profile scan yields ’configuration characteristic’.

Consistent profiles are a unique challenge of global simulations.

Reasonable assumptions about residual pressure are crucial.


Radial electric field effects

W7-X (β = 2%, T∗ = 1 keV)

growth rates

−5 0 5

0.03

0.035

0.04

0.045

0.05

<M>ExB

/ 10−3

γ (

vT/a

)

model A

ion root

model B

frequencies

−5 0 50

0.2

0.4

0.6

0.8

1

1.2

1.4

<M>ExB

/ 10−3

ωm

od

es (

vT/a

)

model A

ion root

model B

Two different radial electric field models used (variation of α):

Physical: Es = αp′iqini⇒ model A

Simple: Es = −α⇒ model B

Asymmetric damping effect


Conclusions

EUTERPE: global (full-volume) gyrokinetic particle-in-cell codefor stellarators.

Hierarchy of models implemented:MHD hybrid, electron fluid, fully gyrokinetic.

New numerical schemes allow faster and more robustelectromagnetic simulations.

Examples:

Fast particle interaction with TAE and internal kink mode.Electrostatic ITG with (quasi-)realistic profiles for stellarators.

Progress, especially for electromagnetic simulations, has beenmade but further testing is necessary.


ITPA benchmark case

International benchmark case for Alfven wave fast-particle interaction.

Circular tokamak A = 10, R = 10 m, B = 3 T, bulk ions: D

ne = 2 · 1019 m−3,Ti = Te = 1 keV

q = 1.71 + 0.16s

Tfast = 400 keV (Maxwellian)

nfast = n0 exp(−∆nLn tanh s−s0

∆n

)with n0 = 0.75 · 1017 m−3,∆n = 0.2, Ln = 0.3, s0 = 0.5

TAE mode


Electromagnetic ITG for W7-X

βe-effect on growth raterelatively weak.

In contrast to tokamak noAITG/KBM branch found inβe scan.

Pullback scheme allows muchfaster (one day) simulationsthan adaptive control variatescheme.

frequency

0 1 2 3 4 5

β [%]

0.0

0.5

1.0

1.5

2.0

ω [

106 s

-1]

adiabatic electronselectromagnetic

pullback

growth rate

0 1 2 3 4 5

β [%]

0.20

0.25

0.30

0.35

0.40

γ [1

06 s-1

]

adiabatic electronselectromagnetic

pullback

⇒ More careful tests necessary to validate schemes.


gyrokinetic simulations for tokamaks and stellarators meeting... · gyrokinetic simulations for...

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